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Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

  • JEFFREY GAITHER (a1), GUY LOUCHARD (a2), STEPHAN WAGNER (a3) and MARK DANIEL WARD (a4)

Abstract

We analyse the first-order asymptotic growth of

\[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \]
The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵ jj for −njn (with j ≠ 0), with ϵ j ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

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Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

  • JEFFREY GAITHER (a1), GUY LOUCHARD (a2), STEPHAN WAGNER (a3) and MARK DANIEL WARD (a4)

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